TITLE: Algorithm to find all (up to isomorphism) perfect matchings of quartic plane graphs
QUESTION [5 upvotes]: I need to find all (up to isomorphism) perfect matchings of some quartic plane graphs. I haven't found any specific algorithm to give me all the perfect matchings. Does anybody know about such an algorithm or any results that could be useful when implementing such an algorithm? At the moment I can only think of a branch-and-bound approach.
I don't really expect there to be an algorithm for this specific case, but I thought I'd mention all properties of the graphs. Maybe there are results for plane graphs, planar graphs or quartic graphs.

REPLY [3 votes]: I want to mention that you can exploit your favorite symbolic algebra package to do this job for smallish graphs. I've used this method successfully for graphs with 20-30 vertices. 
The FKT algorithm expresses the number of matchings of a planar graph as a pfaffian, where each nonzero entry of the matrix is $\pm 1$, corresponding to an edge $e$ of the graph. Write down the same matrix but replace $\pm 1$ by $\pm x(e)$, where $x(e)$ is a formal variable corresponding to the edge $e$, and ask your favorite computer algebra package to take the determinant. The result will be $\sum_M \prod_{e \in M} x(e)$, where the sum is over all matchings $M$. So you can look at the output and immediately read off your matchings.
This is most useful when the graph is bipartite, as the Pfaffian then simplifies to a determinant, and where the graph is small and/or symmetric enough to carry out the orientation part of the algorithm by hand.